The goal of this study is to try to understand multi-strain diseases with the presence of cross-immunity by using mathematical models and other mathematical tools. Cross-immunity occurs when a host who is exposed to one disease, or one strain of a disease, develops resistance or partial resistance to related diseases or strains. It is an important factor in the epidemiology of diseases prone to mutation. This work includes modelling influenza in both presence and absence of controls. It also includes modelling malaria when cross-species immunity is present. In addition, vector-bias of mosquitoes to infected humans is also studied in the single-strain malaria model.

Previous multi-strain mathematical models have elucidated that the degree of cross-protective responses between similar strains, acting as a form of immune selection, generates different behavioural states of the pathogen population. This thesis explores these multi-strain dynamic states, to examine their robustness and stability in the face of pathogenic intrinsic phenotypic variation, and the extrinsic force of immune selection. This is achieved in two main ways: Chapter 2 introduces phenotypic variation in pathogen transmissibility, testing the robustness of a stable pathogen population to the emergence of an introduced strain of higher transmission potential; and Chapter 3 introduces a new model with a possibility of immunity to both strain-specific and cross-strain (conserved) determinants, to investigate how heterogeneity in the specificity of a host immune response alters the pathogen population structure. A final investigation in Chapter 4 develops a method of reverse-pattern oriented modelling using a machine learning algorithm to determine which intrinsic properties of the pathogen, and their combinations, lead to particular disease-like population patterns. This research offers novel techniques to complement previous and ongoing work on multi-strain modelling, with direct applications to a range of infectious agents such as Plasmodium falciparum, influenza A, and rotavirus, but also with a wider potential for other multi-strain systems.

No two species can indefinitely occupy the same ecological niche according to the competitive exclusion principle. When competing strains of the same pathogen invade a homogeneous population, the strain with the largest basic reproductive ratio R0 will force the other strains to extinction. However, over 51 pathogens are documented to have multiple strains [3] coexisting, contrary to the results from homogeneous models. In reality, the world is heterogeneous with the population varying in susceptibility. As such, the study of epidemiology, and hence the problem of pathogen coexistence should entail heterogeneity. Heterogeneous models tend to capture dynamics such as resistance to infection, giving more accurate results of the epidemics. This study will focus on the behavior of multi-pathogen heterogeneous models and will try to answer the question: what are the conditions on the model parameters that lead to pathogen coexistence? The goal is to understand the mechanisms in heterogeneous populations that mediate pathogen coexistence. Using the moment closure method, Fleming et. al. [22] used a two pathogen heterogeneous model (1.9) to show that pathogen coexistence was possible between strains of the baculovirus under certain conditions. In the first part of our study, we consider the same model using the hidden keystone variable (HKV) method. We show that under some conditions, the moment closure method and the HKV method give the same results. We also show that pathogen coexistence is possible for a much wider range of parameters, and give a complete analysis of the model (1.9), and give an explanation for the observed coexistence. The host population (gypsy moth) considered in the model (1.9) has a year life span, and hence, demography was introduced to the model using a discrete time model (1.12). In the second part of our study, we will consider a multi-pathogen compartmental heterogeneous model (3.1) with continuous time demography. We show using a Lyapunov function that pathogen coexistence is possible between multiple strains of the same pathogen. We provide analytical and numerical evidence that multiple strains of the same pathogen can coexist in a heterogeneous population.

Multiple strain path changes during forming lead to complex geometrically necessary dislocation (GND) development in strain gradient fields, inducing internal stresses that contribute to the Bauschinger effect, residual stresses, and springback which alters the final geometrical shape of the part. In order to analyze and design improved processing routes, models must capture the evolution of these internal stresses. However, most models capture the effects of these stresses via phenomenological approaches that require calibration to each new material and strain path. The development of models that capture the underlying physics at the sub-grain level is underway but requires in-depth studies of dislocation behavior (at the relevant meso length scale) in order to guide and validate them. The novel experimental campaign central to this thesis aims to tackle this problem by capturing unprecedented data of dislocation activity for several sheet metals during multiple strain path deformation. The resultant insights provide a new window into multi-path forming of metals, while also aiding the development and validation of two crystal plasticity (CP) models by collaborators at the University of New Hampshire (UNH). The models incorporate internal stresses at the grain and sub-grain levels, respectively. The hardening response due to strain path change during forming of AA6016-T4 was studied at the macro- and micro-level via combined experiments and an elasto-plastic self-consistent (EPSC) model. The experiments demonstrated that possible recombination and/or redirection of dislocations onto different slip systems under strain path change allowed for a gradual elasto-plastic transition, in comparison to a much sharper response upon continued deformation under the same strain path due to buildup and immediate activation of backstresses. The phenomenological backstress law of the EPSC model underpredicted the yield stress response for the strain path change deformations, possibly due to missing sub-grain GND development and an accurate description of associated backstresses. A more detailed experimental study of multi-path deformation for the AA6016-T4 was required in order to guide development of a strain gradient elasto-visco plasticity self-consistent model (SG-EVPSC); the model includes sub-grain strain gradient fields, and related internal stress fields. Total dislocation and GND density were tracked at various points of the deformation, and a complete 3D statistical volume element was characterized, to enable accurate modeling of the microstructure. The tests revealed a relatively lower yield stress response following strain path change, presumably aided by lower latent hardening than self hardening; the tests then showed a rapid accumulation of dislocations on the newly activated slip systems resulting in much higher final dislocation density without affecting the ductility of the pre-strained material. Interestingly, GND development was dominated by the precipitates instead of grain boundaries. These observations are vital for an accurate forming prediction from CPFEA models. Finally, optimized forming conditions of continuous bending under tension produced a ratcheting strain path resulting in a gradual GND development and a more complete retained austenite transformation in quenched-&-partitioned- and TRIP-assisted bainitic ferritic-1180 steels increasing their ductility by at least 360%.

A modelagem epidemiológica é uma importante ferramenta que auxilia os órgãos de saúde no controle de doenças infecciosas, pois permitem analisar e comparar diversas estratégias que facilitam a tomada de decisões e definições de protocolos. A dengue é atualmente a doença viral humana com maior número de casos. Possui índice de mortalidade baixo, entretanto, é endêmica em mais de 100 países e 40% da população mundial está em risco de contrair a infecção. Através dos dados de notificação de dengue no Brasil, evidenciamos que os surtos são sazonais, que há alternância de sorotipos ao longo dos anos e mostramos que a doença é diferente em cada localização, e que somente com uma normalização adequada é possível sugerir um agrupamento coerente de municípios. Neste trabalho, as informações obtidas a partir dos dados são usadas para a estruturação dos modelos matemáticos e para a estimação de parâmetros que validam estes modelos. Comparamos a dinâmica de transmissão de dengue do modelo com um sorotipo, com modelos que permitem a interação de dois, três e quatro sorotipos simultaneamente, além da possibilidade de até quatro infecções sequenciais. Os modelos com múltiplos sorotipos são expandidos do modelo básico que categoriza hospedeiros dentro de uma população como suscetíveis (S), infectados (I) e recuperados (R) e acoplado à dinâmica dos vetores suscetíveis (V) e infectados (Vi). Nossos modelos incluem: um período de imunidade cruzada de forma que o indivíduo adquire imunidade permanente para o sorotipo que já foi infectado e imunidade temporária para os demais; uma forçante de sazonalidade na taxa de nascimento dos vetores; uma assimetria com taxas de transmissão diferentes para cada sorotipo; e o compartimento dos vacinados, com uma vacina tetravalente que confere diferentes imunidades para cada sorotipo. Os resultados mostram que para a reprodução de surtos anuais é necessário a inclusão da forçante de sazonalidade na taxa de nascimento dos vetores, e que o modelo com quatro sorotipos é o que melhor reproduz os dados de incidência de dengue, sendo o mais adequado para analisar estratégias de vacinação com uma vacina tetravalente. Comparamos duas estratégias de vacinação: vacinação aleatória na população e vacinação direcionada para faixas etárias. Neste caso, os resultados demonstram a superioridade da estratégia direcionada e que as escolhas das faixas etárias devem ser definidas por município e não por um protocolo nacional. / Epidemiological modelling is an important tool that assists the health agencies in the control of infectious diseases, since it allows analysing and to compare several strategies that facilitate decision-making and protocol definitions. Dengue is currently the most important vector-borne disease. The mortality rate of dengue is low, however, it is endemic in more than 100 countries and about 40% of the world\'s population is at risk of contracting the infection. Through the dengue notification data in Brazil, we emphasize that the outbreaks are seasonal, there is serotypes alternation over the years and we show that the disease is different in each locality, and that only with a suitable standardization it is possible to propose an appropriate grouping of municipalities. In this work, we use the data information to formulate the mathematical models and for the parameter\'s estimation in order to validate these models. We compare the dynamics of dengue of the one serotype model with the models that allow interaction of two, three and four serotypes simultaneously, including the possibility of at most four sequential infections.The multi-strain models are expanded from the basic model which categorizes the host population as susceptible (S), infected (I), and recovered (R) and coupled with the dynamics of the susceptible (V) and infected (Vi) vectors. Our models include: a period of cross-immunity which means permanent immunity to the serotype of the infection and temporary immunity to the other serotypes; a seasonal forcing in the mosquitoes birth rate; different transmissions rates, so that the models are asymmetric; and the compartment of vaccinated individuals with a tetravalent vaccine which confers different immunities for each serotype. The results show that to reproduce yearly outbreaks it is necessary to include the seasonal forcing in the birth rate of the vectors, and that the four serotypes model is the one that best reproduces the dengue incidence data, being the most suitable model to analyse vaccination strategies with a tetravalent vaccine. We compare two vaccination strategies: random vaccination and vaccination targeted at age groups. In this case, the results demonstrate the superiority of the targeted strategy and that the choices of the age groups should be defined by municipality and not by a national protocol.

A modelagem epidemiológica é uma importante ferramenta que auxilia os órgãos de saúde no controle de doenças infecciosas, pois permitem analisar e comparar diversas estratégias que facilitam a tomada de decisões e definições de protocolos. A dengue é atualmente a doença viral humana com maior número de casos. Possui índice de mortalidade baixo, entretanto, é endêmica em mais de 100 países e 40% da população mundial está em risco de contrair a infecção. Através dos dados de notificação de dengue no Brasil, evidenciamos que os surtos são sazonais, que há alternância de sorotipos ao longo dos anos e mostramos que a doença é diferente em cada localização, e que somente com uma normalização adequada é possível sugerir um agrupamento coerente de municípios. Neste trabalho, as informações obtidas a partir dos dados são usadas para a estruturação dos modelos matemáticos e para a estimação de parâmetros que validam estes modelos. Comparamos a dinâmica de transmissão de dengue do modelo com um sorotipo, com modelos que permitem a interação de dois, três e quatro sorotipos simultaneamente, além da possibilidade de até quatro infecções sequenciais. Os modelos com múltiplos sorotipos são expandidos do modelo básico que categoriza hospedeiros dentro de uma população como suscetíveis (S), infectados (I) e recuperados (R) e acoplado à dinâmica dos vetores suscetíveis (V) e infectados (Vi). Nossos modelos incluem: um período de imunidade cruzada de forma que o indivíduo adquire imunidade permanente para o sorotipo que já foi infectado e imunidade temporária para os demais; uma forçante de sazonalidade na taxa de nascimento dos vetores; uma assimetria com taxas de transmissão diferentes para cada sorotipo; e o compartimento dos vacinados, com uma vacina tetravalente que confere diferentes imunidades para cada sorotipo. Os resultados mostram que para a reprodução de surtos anuais é necessário a inclusão da forçante de sazonalidade na taxa de nascimento dos vetores, e que o modelo com quatro sorotipos é o que melhor reproduz os dados de incidência de dengue, sendo o mais adequado para analisar estratégias de vacinação com uma vacina tetravalente. Comparamos duas estratégias de vacinação: vacinação aleatória na população e vacinação direcionada para faixas etárias. Neste caso, os resultados demonstram a superioridade da estratégia direcionada e que as escolhas das faixas etárias devem ser definidas por município e não por um protocolo nacional. / Epidemiological modelling is an important tool that assists the health agencies in the control of infectious diseases, since it allows analysing and to compare several strategies that facilitate decision-making and protocol definitions. Dengue is currently the most important vector-borne disease. The mortality rate of dengue is low, however, it is endemic in more than 100 countries and about 40% of the world\'s population is at risk of contracting the infection. Through the dengue notification data in Brazil, we emphasize that the outbreaks are seasonal, there is serotypes alternation over the years and we show that the disease is different in each locality, and that only with a suitable standardization it is possible to propose an appropriate grouping of municipalities. In this work, we use the data information to formulate the mathematical models and for the parameter\'s estimation in order to validate these models. We compare the dynamics of dengue of the one serotype model with the models that allow interaction of two, three and four serotypes simultaneously, including the possibility of at most four sequential infections.The multi-strain models are expanded from the basic model which categorizes the host population as susceptible (S), infected (I), and recovered (R) and coupled with the dynamics of the susceptible (V) and infected (Vi) vectors. Our models include: a period of cross-immunity which means permanent immunity to the serotype of the infection and temporary immunity to the other serotypes; a seasonal forcing in the mosquitoes birth rate; different transmissions rates, so that the models are asymmetric; and the compartment of vaccinated individuals with a tetravalent vaccine which confers different immunities for each serotype. The results show that to reproduce yearly outbreaks it is necessary to include the seasonal forcing in the birth rate of the vectors, and that the four serotypes model is the one that best reproduces the dengue incidence data, being the most suitable model to analyse vaccination strategies with a tetravalent vaccine. We compare two vaccination strategies: random vaccination and vaccination targeted at age groups. In this case, the results demonstrate the superiority of the targeted strategy and that the choices of the age groups should be defined by municipality and not by a national protocol.

Dengue fever is endemic in tropical and sub-tropical countries, and some of the important features of Dengue fever spread continue posing challenges for mathematical modelling. We propose a model, namely a system of integro-differential equations, to study a multi-serotype infectious disease. The main purpose is to include and analyse the effect of a general time delay on the model describing the length of the cross immunity protection and the effect of Antibody Dependent Enhancement (ADE), both characteristics of Dengue fever. Analysing the system, we could find the equilibriums in the invariant region. A coexistence endemic equilibrium within the region was proved, even for the asymmetric case. The local stability for the disease free equilibrium and for the boundary endemic equilibriums were proved. We have also results about the stability of the solutions of the system, that is completely determined by the Basic Reproduction Number and by the Invasion Reproduction Number, defined mathematically, as a threshold value for stability. The global dynamics is investigated by constructing suitable Lyapunov functions. Bifurcations structure and the solutions of the system were shown through numerical analysis indicating oscillatory dynamics for specific value of the parameter representing the ADE. The analytical results prove the instability of the coexistence endemic equilibrium, showing complex dynamics. Finally, mortality due to the disease is added to the original system. Analysis and discussions are made for this model as perturbation of the original non-linear system. / A Dengue é endêmica em países tropicais e subtropicais e, algumas das importantes características da dengue continua sendo um desafio para a modelagem da propagação da doença. Assim, propomos um modelo, um sistema de equações integro-diferenciais, com o objetivo de estudar uma doença infecciosa identificada por vários sorotipos. O principal objetivo é incluir e analisar o efeito de um tempo geral de retardo no modelo descrevendo o tempo de imunidade cruzada para a doença e o efeito do Antibody Dependent Enhancement (ADE). Analisando o sistema, encontramos os equilíbrios, onde a existência do equilíbrio de coexistência foi provado, mesmo para o caso assimétrico. A estabilidade local para o equilíbrio livre de doença e para os equilíbrios específicos de cada sorotipo foi provada. Também mostramos resultados para a estabilidade das soluções do sistema que é completamente determinada pelo Número Básico de Reprodução e pelo Número Básico de Invasão, definido matematicamente como um valor limiar para a estabilidade. A dinâmica global é investigada construindo funções de Lyapunov. Adicionalmente, bifurcações e as soluções do sistema foram mostrados via análise numérica indicando dinâmica oscilatória para específicos valores do parâmetro que representa o efeito ADE. Resultados analíticos obtidos pela teoria da perturbação provam a instabilidade do equilíbrio endêmico de coexistência e apontam para um complexo comportamento do sistema. Por fim, a mortalidade causada pela doença é adicionada ao sistema original. Análises e discussões são feitas para este modelo como uma perturbação do sistema não linear original.

In under 6 decades dengue has emerged from South East Asia to become the most widespread arbovirus affecting human populations. Recent dramatic increases in epidemic dengue fever have mainly been attributed to factors such as vector expansion and ongoing ecological, climate and socio-demographic changes. The failure to control the virus in endemic regions and prevent global spread of its mosquito vectors and genetic variants, underlines the urgency to reassess previous research methods, hypotheses and empirical observations. This thesis comprises a set of studies that integrate currently neglected and emerging epidemiological, ecological and evolutionary factors into unified mathematical frameworks, in order to better understand the contemporary population biology of the dengue virus. The observed epidemiological dynamics of dengue are believed to be driven by selective forces emerging from within-host cross-immune reactions during sequential, heterologous infections. However, this hypothesis is mainly supported by modelling approaches that presume all hosts to contribute equally and significantly to the selective effects of cross-immunity both in time and space. In the research presented in this thesis it is shown that the previously proposed effects of cross-immunological reactions are weakened in agent-based modelling approaches, which relax the common deterministic and homogeneous mixing assumptions in host-host and host-pathogen interactions. Crucially, it is shown that within these more detailed models, previously reported universal signatures of dengue's epidemiology and population genetics can be reproduced by demographic and natural stochastic processes alone. While this contrasts with the proposed role of cross-immunity, it presents demographic stochasticity as a parsimonious mechanism that integrates, for the first time, multi-scale features of dengue's population biology. The implications of this research are applicable to many other pathogens, involving challenging new ways of determining the underlying causes of the complex phylodynamics of antigenically diverse pathogens.